A knot is an embedding of the circle (S1) into three-dimensional Euclidean space (R3), or the 3-sphere (S3), since the 3-sphere is compact. Two knots are defined to be equivalent if there is an ambient isotopy between them.
What is meant by knot in physics?
In quantum physics, a knot may be regarded as the orbit in spacetime of a charged particle. One way of calculating the Jones polynomial in quantum theory involves using the Chern-Simons function for gauge fields.
How is knot theory used in physics?
Physical knot theory is used to study how geometric and topological characteristics of filamentary structures, such as magnetic flux tubes, vortex filaments, polymers, DNAs, influence their physical properties and functions.
What does knot mean in math?
In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot).
Why can’t knots have 4 dimensions?
In four dimensions, you can’t tie your shoelaces — because 4D knots don’t work. Any 1D curve in 4D space can be continuously deformed to the unit circle, which is an unknot.
Why do knots exist?
The term knot dates from the 17th century, when sailors measured the speed of their ship using a device called a “common log.” The common log was a rope with knots at regular intervals, attached to a piece of wood shaped like a slice of pie.
Who invented knot theory?
Knots that cannot be so resolved are called prime. The first steps toward a mathematical theory of knots were taken about 1800 by the German mathematician Carl Friedrich Gauss.
Can you have knots in string theory?
Left: In two dimensions, no matter how complicated and convoluted your string is it can never be tied in a knot. Right: Even the simplest knot requires at least three over-under excursions into three dimensional space to get around self-intersections.
Can you have knots in more than 4 dimensions?
Higher-dimensional knots can also be added but there are some differences. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers smooth knots in codimension at least 3.
Is circle a knot?
In fact, a circle is a knot, known as an unknot or a trivial knot because it is so simple. A knot that is not trivial has some sort of folding in space such that no amount of steps can undo the knot into an unknot.
How many knots exist?
The final tally: 352,152,252 knots.
What is a knot diagram?
A knot diagram is a picture of a projection of a knot onto a plane. Usually, only double points are allowed (no more than two points are allowed to be superposed), and the double or crossing points must be “genuine crossings” which transverse in the plane.
Is knot theory pure math?
Knot theory is a branch of pure mathematics, but it is increasingly being applied in a variety of sciences.
When was knot theory created?
In 1867 after observing Scottish physicist Peter Tait’s experiments involving smoke rings, Thomson came to the idea that atoms were knots of swirling vortices in the æther. Chemical elements would thus correspond to knots and links.
What is the study of knots?
Knots and links are studied in topology, which studies properties that are unchanged by continuous transformations. Knots are examples of embeddings, since they are loops living in in 3-dimensional space. A knot is a closed loop of string in three dimensional space.
Do knots exist in higher dimensions?
“For this reason, knotted or linked tubes can’t form in higher-dimension spaces,” said Kephart. The net result is that inflation would have been limited to three dimensions. Additional dimensions, if they exist, would remain infinitesimal in size, far too small for us to perceive.
Can string theory be sheets?
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind as a direct generalization of the world line concept for a point particle in special and general relativity.
Why is speed called knots?
The term knot derives from its former use as a length measure on ships’ log lines, which were used to measure the speed of a ship through the water. Such a line was marked off at intervals by knots tied in the rope.
Why speed is measured in knots?
A nautical mile is slightly more than a standard mile. The term knot dates from the 17th Century, when sailors measured the speed of their ship by the use of a device called a “common log.” This device was a coil of rope with uniformly spaced knots tied in it, attached to a piece of wood shaped like a slice of pie.
Why do planes use knots?
Boats & Planes calculate speed in knots because it is equal to one nautical mile. Nautical miles are used because they are equal to a specific distance measured around the Earth. Since the Earth is circular, the nautical mile allows for the curvature of the Earth and the distance that can be traveled in one minute.
What is a knot expert called?
Knot tying has several commonly used terms. The ancient Latin word for knotting is ‘nodology’, the Greek referred to this art as ‘kompology’. These references are seldom used today, some modern knot tyers prefer the term ‘knottology’ and class themselves as ‘knottologists’.
What is king of all knots?
The bowline is sometimes referred to as King of the knots because of its importance. Along with the sheet bend and the clove hitch, the bowline is often considered one of the most essential knots.
Are all knots Homeomorphic?
So yes all knots are homeomorphic to the circle.
What are the applications of knot theory?
For instance, knot theory is used when modeling DNA and the effects of enzymes on it, as well as in statistical mechanics, when examining the interactions be- tween particles in a system.